Characterization of submartingales of a new class (Σr)

In this work, we shall consider a new class (Σr) of local submartingales of the form Xt = Mt + At, where (Mt)t ⩾ 0 is a càdlàg (right continuous with left limits) local martingale, (At)t ⩾ 0 is a càdlàg increasing process, and the measure (dA) is carried by the set {t: Xt − = 0}. The aim of the present paper is to study the positive and negative parts of processes of this class and establish some martingale characterizations. The formula of relative martingales is derived in terms of last passage time. Finally, by using balayage formula, we calculate predictable compensator.